We introduce the class of \(\alpha \)-firmly nonexpansive and quasi \(\alpha \)-firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where \(\alpha \)-firmly nonexpansive operators coincide with so-called \(\alpha \)-averaged operators. For our more general setting, we show that \(\alpha \)-averaged operators form a subset of \(\alpha \)-firmly nonexpansive operators. We develop some basic calculus rules for (quasi) \(\alpha \)-firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) \(\alpha \)-firmly nonexpansive. Moreover, we will see that quasi \(\alpha \)-firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder’s demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates \(x_{n+1}:=Tx_{n}\) belong to the fixed point set \({{\,\mathrm{Fix}\,}}T\) whenever the operator T is nonexpansive and quasi \(\alpha \)-firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial’s property, then these iterates converge weakly to some element in \({{\,\mathrm{Fix}\,}}T\). Further, the projections \(P_{{{\,\mathrm{Fix}\,}}T}x_n\) converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in \(L_p\), \(p \in (1,\infty ) \backslash \{2\}\) spaces on probability measure spaces.

我们在r一致凸 Banach 空间上引入了类\(\alpha \) -牢固非膨胀和拟\(\alpha \) -牢固非膨胀算子。这扩展了希尔伯特空间的现有概念，其中\(\alpha \) -严格非膨胀算子与所谓的\(\alpha \) -平均算子重合。对于我们更一般的设置，我们证明了\(\alpha \) -平均算子形成了\(\alpha \) -严格非扩张算子的子集。我们为（拟）\(\alpha \) -严格的非膨胀算子开发了一些基本的微积分规则。特别是，我们证明了它们的组合和凸组合再次（准）\(\alpha \) -坚决不膨胀。此外，我们将看到拟\(\alpha \) -牢固非膨胀算子享有渐近正则性。然后，基于 Browder 的半封闭性原理，我们证明对于r -一致凸 Banach 空间，迭代的弱聚类点\(x_{n+1}:=Tx_{n}\)属于不动点集\({ {\,\mathrm{Fix}\,}}T\)每当算子T是非膨胀的并且准\(\alpha \)时。如果另外空间具有 Fréchet 可微范数或满足 Opial 的性质，则这些迭代弱收敛到\({{\,\mathrm{Fix}\,}}T\) 中的某个元素。此外，预测\(P_{{{\,\mathrm{Fix}\,}}T}x_n\)强烈收敛到这个弱极限点。最后，我们给出了三个可以应用我们的理论的说明性例子，即来自无限维神经网络、半群理论和\(L_p\) 中的收缩投影，\(p \in (1,\infty ) \backslash \{ 2\}\)概率度量空间上的空间。